Categorifies Hecke algebra subalgebra using sheaves and bimodules.
problem Categorify maximal commutative subalgebra of type A Hecke algebra.
method Construct monoidal functor from coherent sheaves on flag Hilbert scheme to Soergel bimodules, match Hochschild homology with sheaf Euler characteristics.
result Categorified projectors correspond to dg algebras of functions on affine charts of flag Hilbert schemes, conjecturally matching homology of glN projectors. Knot homology of Coxeter links identified with line bundles on Hilbert schemes.
problem Identifying knot homology for Coxeter links.
method Using line bundles on a generalized flag Hilbert scheme of points in \(\mathbb{C}^2\).
result Knot homology of Coxeter links corresponds to sections of a specific line bundle.
Generalizes soft noncommutative schemes to flag varieties.
problem Applying soft noncommutative schemes to flag varieties.
method Generalization via toric geometry and distinguished affine charts.
result Soft noncommutative schemes can be applied to flag varieties.
The study connects Hilbert entropy to non-differentiability points of limit sets in flag spaces.
problem Understanding non-differentiability points in limit sets of convex projective structures.
method Introduces hyperplane conicality for θ-Anosov representations and uses it to prove properties of boundary maps. result Hilbert entropy is linked to the Hausdorff dimension of non-differentiability points in flag spaces.
This paper solves Hilbert's fourth problem for constant curvature metrics.
problem Classifying metric geometries with shortest straight lines in constant curvature settings.
method Analyzing Finsler manifolds with constant flag curvature, deriving distance formulas, and proving global geometry theorems.
result Complete characterization of global geometry for constant flag curvature metrics.
We construct complexes P1n of Soergel bimodules which categorify the Young idempotents corresponding to one-column partitions. A beautiful recent conjecture of Gorsky-Rasmussen relates the Hochschild homology of categorified Young idempotents with the flag Hilbert scheme. We prove this conjecture for P1n an…
It is well known that a system of homogeneous second-order ordinary differential equations (spray) is necessarily isotropic in order to be metrizable by a Finsler function of scalar flag curvature. In Theorem 3.1 we show that the isotropy condition, together with three other conditions on the Jacobi endomorphism, chara…
The paper introduces a new spray deformation and finds new metrics.
problem Characterizing metrizability of new spray structures.
method Introducing one form deformation of sprays and analyzing their metrizability.
result Obtained new projectively flat metrics of constant flag curvature 1.
Solves geodesics and Laplace-Beltrami spectrum on flag manifolds.
problem Geodesics and Laplace-Beltrami spectrum on flag manifolds.
method Invariant metrics and finite-dimensional approximations.
result Explicit solutions for geodesics and spectrum.
Characterizes Hilbert schemes and their geometric properties.
problem Understanding transverse Hilbert schemes and their geometric properties.
method Characterization through bi-Poisson structures and hyperkähler geometry.
result Characterization of transverse Hilbert schemes and description of their hyperkähler geometry.
For spherical Tits buildings of the classical types there are well-known explicit descriptions as flag complexes. Similarly for affine buildings of the classical types there are explicit constructions in terms of lattices. In this article we generalize the flag complex description to twin cities, a generalization of tw…
New systems derived from Hilbert schemes on surfaces.
problem Characterizing completely integrable systems on surfaces.
method Using transverse Hilbert schemes to associate systems to surfaces.
result Characterized completely integrable systems arising from Hilbert schemes.
We give a new proof of the Jantzen sum formula for integral representations of Chevalley schemes over Spec Z. This is done by applying the fixed point formula of Lefschetz type in Arakelov geometry to generalized flag varieties. Our proof involves the computation of the equivariant Ray-Singer torsion for all equivarian…
In this paper we study domains in flag manifolds which are bounded in an affine chart and whose projective automorphism group acts co-compactly. In contrast to the many examples in real projective space, we will show that no examples exist in many flag manifolds. Moreover, in the cases where such domains can exist, we …
Study the geometry of curves in projective spaces.
problem Understand the natural geometry of curves in projective spaces.
method Describe the geometry of Hilbert schemes of curves.
result Describe the geometry of Hilbert schemes of curves in projective spaces.
The study connects polygon areas and projective structures in 3D space.
problem Relating polygon areas and projective structures in 3D space.
method Investigates positive tuples of complete flags in R^3 and their associated polygons in RP^2.
result Establishes a relationship between Holmes-Thompson area and projective structures.
A unified approach to geometric, symbol and deformation quantizations on a generalized flag manifold endowed with an invariant pseudo-Kaehler structure is proposed. The Hilbert space of states is realized via the Bott-Borel-Weil theorem in the sheaf cohomology of the geometric quantization line bundle. The correspondin…
In 1929, Paul Funk and Ludwig Berwald gave a characterization of Hilbert geometries from the Finslerian viewpoint. They showed that a smooth Finsler metric in a convex bounded domain of Rn is the Hilbert geometry in that domain if and only if it is complete, if its geodesics are straight lines and if its fl…
Study of holomorphic triples on surfaces leads to Vafa-Witten invariants.
problem Understanding moduli spaces of holomorphic triples on surfaces.
method Studied moduli space of holomorphic triples with Schmitt stability condition, observing perfect deformation-obstruction theory for large stability parameters.
result Connection between higher rank flag sheaves and Vafa-Witten theory on threefolds.
Extends gl(m|k) construction using Hilbert scheme of points.
problem Quantum invariants of super-algebras gl(m|k).
method Using Hilbert scheme of points on C^2.
result Constructs triply graded link homology for gl(m|k).
Study of Hilbert schemes and Coulomb branches of hypertoric varieties.
problem Understanding the geometry and topology of Coulomb branches of hypertoric varieties.
method Investigation of transverse equivariant Hilbert schemes and Hamiltonian reductions, proposing new metrics.
result Coulomb branches of hypertoric varieties can be constructed as Hilbert schemes or Hamiltonian reductions.
New knot homology uses sheaves on Hilbert schemes.
problem Developing knot invariants using sheaves on Hilbert schemes.
method Constructing 2-periodic complexes of sheaves on Hilbert schemes and computing their cohomology.
result The cohomology of these complexes gives knot invariants.
In 2001, Zhongmin Shen asked if it is possible for two projectively related Finsler metrics to have the same Riemann curvature tensor, [14, page 184]. In this paper, we provide an answer to this question, within the class of Finsler metrics of scalar flag curvature. In Theorem 3.1, we show that the answer is negative, …
The paper connects algebraic geometry to threefold theories.
problem Enumerative geometry of nested Hilbert schemes.
method Study of threefold theories including Donaldson-Thomas, Vafa-Witten, and Seiberg-Witten.
result Connections between algebraic geometry and threefold theories.
New invariant restores symmetry in link homology and matches Hilbert scheme ideals.
problem Restoring missing symmetry in Khovanov-Rozansky homology.
method Defining a deformation of Khovanov-Rozansky homology with parameters yc. result Matches ideals in Haiman's description of isospectral Hilbert scheme.
New representation of curves helps prove complex geometry result.
problem Understanding cohomologically stable curves in projective space.
method Using commuting matrix polynomials to represent curves and show isomorphism to hyperkähler quotient.
result Hilbert scheme isomorphic to a hyperkähler quotient.
Theory developed for Hilbert geometry over valued fields, linking real and non-Archimedean geometries.
problem Understanding Hilbert geometry over general valued fields and their limits.
method Developed a theory of Hilbert geometry over general ordered valued fields, proving ultralimit results.
result Ultralimit of rescaled real Hilbert geometries is isometric to a non-Archimedean Hilbert metric space.
Study shows convergence of cscK surfaces in Hilbert scheme.
problem Understanding convergence of cscK surfaces.
method Gromov--Hausdorff convergence and Hilbert scheme approach.
result Established convergence of non-collapsed polarized cscK surfaces in a Hilbert scheme.
The paper analyzes Tikhonov regularization in Hilbert scales for statistical inverse problems.
problem Statistical inverse problems in Hilbert scales with general noise.
method Tikhonov regularization scheme with conditional stability estimates and high probability error bounds.
result Explicit rates of convergence for oversmoothing and regular cases over defined regularity classes.
We conjecture an expression for the dimensions of the Khovanov-Rozansky HOMFLY homology groups of the link of a plane curve singularity in terms of the weight polynomials of Hilbert schemes of points scheme-theoretically supported on the singularity. The conjecture specializes to our previous conjecture relating the HO…
We study the partial resolutions of singularities related to Hilbert schemes of points on an affine space. Consider a quotient of a vector space V by an action of a finite group G of linear transforms. Under some additional assumptions, we prove that the partial desingularization of Hilbert type is smooth only if t…
Lectures on link homology and its algebraic/geometric models.
problem Defining and understanding link homology.
method Definition and properties of Khovanov-Rozansky triply graded homology, and three geometric models.
result Three geometric models for link homology: braid varieties, Hilbert schemes, and affine Springer fibers.
3D topological models link to HOMFLYPT homology via braids.
problem Understanding topological invariants of links and braids.
method 3D topological B-models with Hilbert schemes of points.
result Hilbert space of braids corresponds to HOMFLYPT homology.
New geometric structures on surfaces generalize complex and real Lie algebra properties.
problem Generalizing geometric structures associated with Lie algebras.
method Define and analyze generalizations of punctual Hilbert schemes for complex and real Lie algebras.
result Construct geometric structures homeomorphic to Hitchin components.
An unobstructedness theorem is proved for deformations of compact holomorphic Poisson manifolds and applied to a class of examples. These include certain rational surfaces and Hilbert schemes of points on Poisson surfaces. We study in particular the Hilbert schemes of the projective plane and show that a generic deform…
The intersection of a complex plane curve with a small three-sphere surrounding one of its singularities is a non-trivial link. The refined punctual Hilbert schemes of the singularity parameterize subschemes supported at the singular point of fixed length and whose defining ideals have a fixed number of generators. We …
New geometric construction of spectral sequences from Khovanov homology.
problem Understanding spectral sequences from Khovanov homology.
method Cylindrical model to compute Fukaya categories of Hilbert schemes.
result New geometric construction of spectral sequences from Khovanov homology.
The well-known Funk metric F(x,y) is projectively flat with constant flag curvature K=-1/4 and the Hilbert metric H(x,y):=(F(x,y)+F(x,-y))/2 is projectively flat with constant curvature K=-1. These metrics are the special solutions to Hilbert's Fourth Problem. In this paper, we construct a non-trivial R-flat spray usin…
Geometrically computes sheaves linking HOMFLY-PT homology to Hilbert schemes.
problem Linking HOMFLY-PT homology to geometric structures on Hilbert schemes.
method Geometric sheaf theory, Hochschild homology formality, Hilbert schemes of points.
result Established formalism connecting HOMFLY-PT homology to coherent sheaves on Hilbert schemes.
New method computes triply graded link homology for torus links.
problem Computing triply graded link homology for torus links.
method Introducing a new method for triply graded link homology specifically adapted to torus links.
result Exact answers for (n,n)-torus links, verifying conjectures about homology and Hilbert schemes.
We show that the regular Slodowy slice to the sum of two semisimple adjoint orbits of GL(n,C) is isomorphic to the deformation of the D2-singularity if n=2, the Dancer deformation of the double cover of the Atiyah-Hitchin manifold if n=3, and to the Atiyah-Hitchin manifold itself if n=4. For higher n, such…
Aganagic and Shakirov propose a refinement of the SU(N) Chern-Simons theory for links in three manifolds with S^1-symmetry, such as torus knots in S^3, based on deformation of the S and T matrices, originally found by Kirillov and Cherednik. We relate the large N limit of the S matrix to the Hilbert schemes of points o…
Constructs a new geometric structure on surfaces to generalize Teichmüller theory.
problem Exploring new geometric structures in Teichmüller theory.
method Uses the punctual Hilbert scheme of the plane to construct a higher complex structure and explores its properties.
result Establishes a canonical diffeomorphism between the moduli space of higher complex structures and Hitchin's component.
New geometric structure on surfaces generalizing complex structures.
problem Defining and analyzing new geometric structures on surfaces.
method Using the punctual Hilbert scheme of the plane to define higher complex structures.
result Moduli space of higher complex structures is a generalization of Teichmüller space and conjecturally isomorphic to Hitchin's component.
The paper defines Dirichlet domains for Anosov subgroups in Lie groups.
problem Finding finite-sided Dirichlet domains for Anosov subgroups in Lie groups.
method Introducing a sufficient condition for finite-sided Dirichlet domains in semisimple Lie groups for polyhedral Finsler metrics.
result Finitely generated subgroups of semisimple Lie groups can have finite-sided Dirichlet domains under certain conditions.
New examples of k-regular maps to Grassmannians found via algebraic geometry.
problem Constructing k-regular maps to Grassmannians.
method Algebraic geometry, following Buczyński-Januszkiewicz-Jelisiejew-Michałek.
result First examples of k-regular maps for τ ≥ 2.
Constructs k-regular maps using algebraic geometry.
problem Determine the minimal value of N for k-regular maps from R^m to R^N.
method Algebraic geometry methods to construct k-regular maps and relate upper bounds to the dimension of Gorenstein schemes.
result Explicit examples and upper bounds for k-regular maps for k<6 and arbitrary m and k.
Researchers develop Orlov-Schulman symmetries for self-dual conformal structures.
problem Developing symmetries for self-dual conformal structures.
method Explicit proof of compatibility with Lax-Sato flows, dressing scheme based on Riemann-Hilbert problem.
result Construction and proof of compatibility of Orlov-Schulman symmetries.